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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p><dfn class="terminology">Diagonalization</dfn>Consider</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\bf x}^{\prime}={\bf A}\,{\bf x}.
\end{equation*}
</div>
<p class="continuation">If <span class="process-math">\({\bf A}\)</span> is diagonal, i.e. </p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\bf x}^{\prime}=\left(
\begin{array}{cccc}
a_{11} &amp; 0 &amp; \cdots &amp; 0\\
0 &amp; a_{22} &amp; \cdots &amp; 0\\
\vdots &amp; \vdots &amp; \ddots &amp; \vdots\\
0 &amp; 0 &amp; \cdots &amp;  a_{nn}
\end{array}
\right) {\bf x},
\end{equation*}
</div>
<p class="continuation">then</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\begin{array}{c}
x_1^{\prime}=a_{11} x_1,\\
x_2^{\prime}=a_{22} x_2,\\
\vdots\\
x_n^{\prime}=a_{nn} x_n,\\
\end{array}
\end{equation*}
</div>
<p class="continuation">and these are decoupled equations for <span class="process-math">\(x_1, x_2, \cdots, x_n\text{.}\)</span> The solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\begin{array}{c}
x_1=C_1 e^{a_{11} t},\\
x_2=C_2 e^{a_{22} t},\\
\vdots\\
x_n=C_n e^{a_{nn} t}.
\end{array}
\end{equation*}
</div>
<span class="incontext"><a href="sec6_5.html#p-280" class="internal">in-context</a></span>
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